\section*{Aufgabe 1}
Gegeben: \begin{itemize}
			\item $\Pp = \{ x,y,z\}$ 
			\item $\C = \{ 1,2,3 \}$			
			\item $Pr(x) = \dfrac{1}{4}, Pr(y) = \dfrac{1}{4}, Pr(z) = \dfrac{1}{2}$
		 \end{itemize}
Bestimme: $Pr(m|c)$ für alle $m \in \Pp$ und alle $c \in \C$ \\ [0.2 cm]
		  $Pr(x|1) = \dfrac{Pr(x \cap 1)}{Pr(1)} = \dfrac{Pr(x) \cdot Pr(1)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{4} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
		   = \dfrac{\dfrac{1}{12}}{\dfrac{1}{3}} = \dfrac{3}{12} = \dfrac{1}{4}$ \\
		  $Pr(x|2) = \dfrac{Pr(x \cap 2)}{Pr(2)} = \dfrac{Pr(x) \cdot Pr(2)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{4} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
		   = \dfrac{\dfrac{1}{12}}{\dfrac{1}{3}} = \dfrac{3}{12} = \dfrac{1}{4}$ \\
  		  $Pr(x|3) = \dfrac{Pr(x \cap 3)}{Pr(3)} = \dfrac{Pr(x) \cdot Pr(3)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{4} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{12}}{\dfrac{1}{3}} = \dfrac{3}{12} = \dfrac{1}{4}$ \\
  		  $Pr(y|1) = \dfrac{Pr(y \cap 1)}{Pr(1)} = \dfrac{Pr(y) \cdot Pr(1)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{4} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{12}}{\dfrac{1}{3}} = \dfrac{3}{12} = \dfrac{1}{4}$ \\
  		  $Pr(y|2) = \dfrac{Pr(y \cap 2)}{Pr(2)} = \dfrac{Pr(y) \cdot Pr(2)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{4} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{12}}{\dfrac{1}{3}} = \dfrac{3}{12} = \dfrac{1}{4}$ \\
  		  $Pr(y|3) = \dfrac{Pr(y \cap 3)}{Pr(3)} = \dfrac{Pr(y) \cdot Pr(3)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{4} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{12}}{\dfrac{1}{3}} = \dfrac{3}{12} = \dfrac{1}{4}$ \\
  		  $Pr(z|1) = \dfrac{Pr(z \cap 1)}{Pr(1)} = \dfrac{Pr(z) \cdot Pr(1)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{2} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{6}}{\dfrac{1}{3}} = \dfrac{3}{6} = \dfrac{1}{2}$ \\
  		  $Pr(z|2) = \dfrac{Pr(z \cap 2)}{Pr(2)} = \dfrac{Pr(z) \cdot Pr(2)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{2} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{6}}{\dfrac{1}{3}} = \dfrac{3}{6} = \dfrac{1}{2}$ \\
  		  $Pr(z|3) = \dfrac{Pr(z \cap 3)}{Pr(3)} = \dfrac{Pr(z) \cdot Pr(3)}{\dfrac{1}{3}} = \dfrac{\dfrac{1}{2} \cdot \dfrac{1}{3}}{\dfrac{1}{3}}
  		   = \dfrac{\dfrac{1}{6}}{\dfrac{1}{3}} = \dfrac{3}{6} = \dfrac{1}{2}$ \\ [0.2 cm]		   
		   
Das Verfahren ist perfekt geheim, da gilt: \\
$Pr(m|c) = Pr(m)$ für alle $m \in \Pp$ und alle $c \in \C$